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]]>Regarding stories: I think this is hard in olympiad culture now because people very rarely talk about how they come up with problems in the first place (you’re probably the main exception to this). I think it’s partly due to a disconnect between the problem authors and the conestants (the former are generally much older). I agree it would be really cool if stories became a more common thing.

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]]>For what it’s worth I haven’t seen any of this before, but even if I had I’m sure that other readers of the blog would have appreciated it. So thank you again for sharing.

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]]>Thanks for an excellent post, as always. What you’re describing has been studying extensively in cognitive psychology under the umbrella-term “acquisition of expertise”. It may very well be that you are extremely familiar with everything I’m about to write, but just in case you didn’t have the chance to look into it I’d thought I’d throw a few links at you. For example, you write:

“The status quo is that people do bucket sorts based on the particular technical details which are present in the problem.”

One of the most famous studies in the field is Chi, Feltovich, Glaser “Categorization and representation of physics problems by experts and novices.” [1] They asked PhD’s, Profs, and undergraduate to solve problems in classical mechanics, and tried to probe their thought-processes. One of their interesting finds is that undergraduates tend to categorize problems based on the superficial details (the existence of springs, inclined-planes, etc.), whereas experts tend to categorize problems based on the physical principle they expect to use in the solutions (Newton’s second law etc.).

You wrote: “I realized that had the students just ignored the task “prove {n \le 3^k}” and spent some time getting a better understanding of the {\varphi} structure, they would have had a much better chance at solving the problem.”

The same study also found that experts tend to spend more time at the beginning, evaluating the problem (trying to decide “what kind” it is); whereas the beginning undergraduates tend to jump into the fray, immediately trying to solve the problem with some equations.

Finally, you say that you “spend a lot of time trying to classify the main ideas into categories or themes,” and advocate “to keep a journal or blog of the problems you’ve done.” This is part of the process of chunking [2], and the thinking is that experts differ from non-experts in the complexity and quality of their chunks (one could chunk chunks, and iterating this is the “complexity”). In fact, the Russian school of chess ~30 years ago asked students to collect favourite positions on flash-cards, and have their analysis on the other side. The positions were supposed to be examples of typical situations/ideas/themes, in the same way you’re advocating keeping a journal of typical problems. [I don’t have a reference for this; I believe I’ve read it in a Jermey Silman book, but I’ll have to dig a little bit to find it.] Since chess can be easily separated into levels of expertise (based on the ELO rating), they are the subjects of a lot of chunking studies [Adriaan de Groot should be the first link in this rabbit-hole].

My apologies again if this is all old-hat to you =)

[1] http://onlinelibrary.wiley.com/doi/10.1207/s15516709cog0502_2/pdf

[2] https://en.wikipedia.org/wiki/Chunking_(psychology)

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]]>Consider the following problem: 12% of a sphere is pained black. Prove there exists a cube inscribed in that sphere with all 8 white vertices.

It can be solved via straightforward application of the probabilistic method: Inscribe a cube in the sphere at random. The indicator random variable of whether or not each vertex is black has expectation 0.12 so the expected number of black vertices is, by linearity of expectation, 0.96, so there must exist an inscribed cube with 0 black vertices or all 8 white vertices. The shortness of this solution makes it suitable for making the discussion very concrete.

That being said, coming up with the above solution seems to be hard, unless one is very good at the probabilistic method, as I believe, if i recall correctly, that in Arthur Engel’s PSS, where the weaker problem for a rectangular prism is given, he states he doesn’t even know if the result is true for a cube. I had the pleasure of coming up with this solution when I was sufficiently unfamiliar with the probabilistic method to have to think hard about this problem. According to what you said in the previous article, this suggests that storing this problem as a cherished memory (which I do) should help me solve other problems which have no obvious relation to this one (problems which are not just one application of the probabilistic method, as I didn’t perceive this problem like that initially either) but which are connected to this problem via a “philosophy” of “what does this problem feel like”.

I’m not sure if my initial motivation for this proof matches your “global” (and it wouldn’t be a perfect example because I had at least SEEN the probabilistic method before) but your description of “global” seems to include this problem, and enable one to solve it without having ever seen probabilistic method as long as they recall linearity of expectation and the fact that if the expected number of black vertices is 0.96, there must be a cube with 0 black vertices.

Is this problem “global” (by the way no pun intended as the problem is about a sphere)?

If so: Could you please explain how thinking about this problem in terms of your “global” would motivate the solution I gave (if we assume the solver knows the material and your global philosophy but either doesn’t know probabilistic method or cannot simply think “oh this is just probabilistic method”/”this is similar to tons of other probabilistic method problems I’ve done” – because my understanding is probabilistic method is a technical theme, and “global” is a broader heuristic)? What about the problem, to somebody who doesn’t realize probabilistic method is fundamentally suited to it, suggests it is a “global” problem?

Thank you

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]]>[BTW, regarding windmill in particular, you may also be interested in https://artofproblemsolving.com/community/c6h19770 (see my post for spoiler).]

Also, by Global you seem to really mean something like Local-to-Global (piecing together local info) as opposed to, say, leading order asymptotic analysis, though I may be wrong.

Cheers,

V

P.S. Thematic heuristics and intuition are definitely good, but I would also like to see more emphasis on *story* and *curiosity* in Olympiad culture—what/why things are interesting and how to ask better questions—in which case technique can play a helpful organizational role. In Expii (esp. Solve) problem tagging we often use both concrete topic tags and meta ones (often from https://www.expii.com/map/3624 or https://www.expii.com/map/3670 for instance), so that one could learn from either lens in principle (though meta is not a current priority for practical reasons).

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